Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(11,15)(13,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,34)(35,40)(36,41)(37,42)(38,39)(43,44)(45,47)(46,48);(3,7)(6,8)(9,10,12,14)(11,13,15,16)(18,19)(20,27)(24,26)(28,32)(34,39)(36,42)(40,47)(43,46);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,35,38,45)(34,40,39,47)(36,46,42,43)(37,44,41,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36,38,42)(34,41,39,37)(35,43,45,46)(40,44,47,48);(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,41)(34,42,39,36)(35,44,45,48)(40,43,47,46)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=f,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4371123);